1. Miniconference on the Mathematics of Computation
MTH 210
Number theory
Dr. Anthony Bonato
Ryerson University

2. Number theory study of numbers, usually, integers primes
later: congruences and Diophantine equations

3. Divisors an integer m is a divisor of n if n = pm for some integer p
examples:
divisors of 10: 1, 2, 5, 10
divisors of 16: 1, 2, 4, 8, 16
divisors of 19: 1, 19

4. Primes 19 is a special number
only divisors are 1, 19
integer n > 1 is prime if its only divisors are 1 and n
NB: 1 is not prime

5. Primes up to 100
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97…

6. Primes never end
Key fact: there are infinitely many primes.

7. Other kinds of numbers rationals: p/q, p,q integers
reals: numbers with a decimal expansion
complex numbers: a + bi, where a, b reals, i2 = -1

8. Irrational numbers a real number is irrational if it is not rational
that is, it isn’t a fraction of integers

9. An irrational number
Key fact: 2 is irrational.

10. Other irrationals irrational, but much tougher to prove: π, e
no one knows if π + e is irrational!

11. Exercises